Question

If `vec"a"` is any non-zero vector, then `(vec"a" .hat"i")hat"i" + (vec"a".hat"j")hat"j" + (vec"a".hat"k")hat"k"` equals ______.

Answer 1

If `vec"a"` is any non-zero vector, then `(vec"a" .hat"i")hat"i" + (vec"a".hat"j")hat"j" + (vec"a".hat"k")hat"k"` equals `vec"a"`.

Explanation:

Let `vec"a" = "a"_1hat"i" + "a"_2hat"j" + "a"_3hat"k"`

∴ `vec"a"*hat"i" = ("a"_1hat"i" + "a"_2hat"j" + "a"_3hat"k") * hat"i"`

Similarly, `vec"a" * hat"j" = "a"_2` and `vec"a" * hat"k" = "a"_3`

∴ `(vec"a" * hat"i")*hat"i" + (vec"a" * hat"j")hat"j" + (vec"a" * hat"k")*hat"k" = "a"_1hat"i" + "a"_2hat"j" + "a"_3hat"k" = vec"a"`

Hence, the value of the filler is `vec"a"`.